The radius of the first orbit of H-atom is r. then the radius of the first orbit of `Li^(2+)` will be:

Options

A. `(r)/(9)`
B. `(r)/(3)`
C. `3r`
D. 9r

Explanation

To find the radius of the first orbit of the Li²⁺ ion, we can use the formula for the radius of an electron orbit in a hydrogen-like atom:

\[ r = r_0 \cdot \frac{n^2}{Z} \]

Where:
- \( r \) is the radius of the orbit,
- \( r_0 \) is a constant (the radius of the first orbit of hydrogen),
- \( n \) is the principal quantum number (1 for the first orbit),
- \( Z \) is the atomic number of the element.

### Step-by-Step Solution:

1. **Identify the radius for the hydrogen atom (H)**:
- For hydrogen (H), the atomic number \( Z = 1 \) and the principal quantum number \( n = 1 \).
- Therefore, the radius of the first orbit of hydrogen is:
\[ r_H = r_0 \cdot \frac{1^2}{1} = r_0 \]

2. **Identify the radius for the lithium ion (Li²⁺)**:
- For lithium (Li²⁺), the atomic number \( Z = 3 \) and the principal quantum number \( n = 1 \) (since we are looking for the first orbit).
- Therefore, the radius of the first orbit of lithium ion is:
\[ r_{Li^{2+}} = r_0 \cdot \frac{1^2}{3} = \frac{r_0}{3} \]

3. **Relate the two radii**:
- We have \( r_H = r_0 \) and \( r_{Li^{2+}} = \frac{r_0}{3} \).
- Thus, the radius of the first orbit of Li²⁺ in terms of the radius of the first orbit of H is:
\[ r_{Li^{2+}} = \frac{r_H}{3} \]

### Final Answer:
The radius of the first orbit of Li²⁺ will be \( \frac{r}{3} \).